Two Topics in Chiral Effective Lagrangians

In the absence of nucleons, we use partial wave unitarity, to show that the chiral expansion parameter must be close to $p^2\mathrm{/}4\pi f_{\pi }^2$ rather than $p^2\mathrm{/}16{\pi }^2{f}_{\pi }^2$ as previously suggested, where $p$ is a typical pion momentum and $f_{\pi }$ the pion decay constant. When nucleons are included, we apply the Tani-Foldy-Wouthuysen (TFW) transformation to the pion-nucleon effective Lagrangian to obtain an expansion in powers of $1\mathrm{/}{m}_N$ (inverse nucleon mass). The results are presented up to order $\mathcal{O}(1\mathrm{/}{m}_N^3)$, corresponding to $\mathcal{O}(p^4)$ in the momentum. In this case partial wave-unitarity is also lost in about the same range of momenta.